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nice image,
like a peaceful day at the ocean beach with mild warm breeze and mellow surf

sweet and mellow

mellow yellow
mellow blue
mellow reds need no meds
and or golden Hue-man too!


I am beat and taking a couple days off.
just finished a 70 page pdf
started maybe ...a week ago ...I cannot remember now..
....editing is insane...

The typos and such are like quantum foo boogers,
you gotta pick em out and try and keep a clean nose hole.

Certainly not like the late 70s and all that Billy Mays Oxycrack coca mocha mania.

we need a Rip van Winkle emoticon

or that Jewish harmonic code creepo in the Pi movie.

[Image: pi_movie_film_aronofsky.jpg]


see you soon
yea baby,
a little time to enjoy before crashing.
No marathon mathemamia for awhile!


This is a Lyapunov experiment,
with new methods of replication.

[Image: spVFt.jpg]
I am still fascinated with this grid work.
Here I have made into ....E9 Grid Construct.
The actual grid I could have gotten to about 10000 by 10000 pixels
but my computer seems to limit out at around 7000 by 7000.
So this is the reduced grid 1500 by 1500
a close up of the interior.
Enneagon geometry.
and by the ways, it is definitely the image magnification program
that Keith has on this forum which causes the image distortion seen,
as the images are far sharper and less wiggly here....

[Image: YHYB7.jpg]

[Image: VW2RR.jpg]
...the Lyapunovs you rendered look like lacquered, gold inlaid wood...
like rare Chinese tabletops!!!

[Image: chinese-checkers-oriental-lid.jpg]

OK...I tried to do something with Lyapunov curves.

I did find out something I've never seen before.
I was researching the  Lyapunov "sequence" parameter...
Read this...
The link shows more examples of what happens as the sequence string becomes ever more complex.
Quote:A Zoom in Lyapunov Space for the Sequence {ab}
[Image: lyapunov-strip.gif]
Click on the image strip to load the QuickTime™ movie.

If we let "a" and "b" be the axis in some parameter space (call it the Lyapunov space) we can then calculate the exponent for all allowable combinations of parameters...
The most impressive feature of these diagrams is their three-dimensional appearance. The swallow-tail structure in the last picture, when rendered in more detail, looks like a solid blob and the superstable arms appear to cross over one other. When the sequence is reversed to {ba} the crossing of the arms reverses. At these locations, we have the coexistence of two attractors, each of which may have a different period. Markus found extremely high levels of basin interleaving in more complicated sequences. The ultimate fate of an orbit (that is, to which basin it will attract) depends on which parameter value we start with. There is no analog of this behavior in the driven harmonic oscillator (although this does not preclude such behavior from being found in other continuous systems).
We have uncovered a new type of phenomena.

Anyway, I zoomed way out and found that these curves are part of a greater Escheresque grid which, as far as I can see, extends to infinity......

[Image: Lyapunov6.jpg]
Quote:There is no analog of this behavior in the driven harmonic oscillator,
although this does not preclude such behavior from being found in other continuous systems,
We have uncovered a new type of phenomena.

That is absolutely wild!

What do you mean by "zoomed out" ?

From what exactly?

Quote:are part of a greater Escheresque grid which, as far as I can see, extends to infinity......

All replicating geometries extend to infinity.
What you have there is this basically:
an expanding Lyapunov Tile.
You should be able to extract a single rectangular component of it all.
You might have to rotate the image
a certain amount of degrees to computer accomplish the precise splice.

I think that is what they are saying.....dunno for sure,
but unexpected replicating pattern..,
but they are correct in this:

Quote:(although this does not preclude such behavior from being found in other continuous systems).
We have uncovered a new type of phenomena.

Lyapunov fractals sometimes get a monotonous but some of them are just fabulous.
which remonds me
I have an image using a Lyapunov fractal from wayyyyyy back,
I will endeavor to find it.

found it, and I reduced size so it was palatable to view.
Center is a section of a Lyapunov fractal in replication.

Try and center the image first. It looks great on a 19 inch screen or bigger

Skyburst of Love

[Image: gkLT7.jpg]
That Lyapunov Tile replication had me also go wayyy back
and find one of my really elaborate Pentad Tiles.
This one was blurry unfortunately, but it still made it into a penta-grid.
It is amazing what a single replicating tile can do.
In this case I reconstructed pieces of the Pentad Octagon ...3 ways left from Tuesday.
I couldn't even begin to remember how that one was made,
but it has little square root numbers all through it from the original Pentad Octagon design,
because I forgot to remove them wayyyy back when

Take the image into your file and reverse the color!

scroll very slowly and pick out smaller areas and view them as you go down,
them come back up slowly
will give the best perspective.

[Image: cr0So.jpg] ... .html?_r=1

Benoit Mandelbrot, Mathematician, Dies at 85

Benoît B. Mandelbrot, a maverick mathematician who developed an innovative theory of roughness and applied it to physics, biology, finance and many other fields, died on Thursday in Cambridge, Mass. He was 85.

His death was caused by pancreatic cancer, his wife, Aliette, said. He had lived in Cambridge.

Dr. Mandelbrot coined the term “fractal” to refer to a new class of mathematical shapes whose uneven contours could mimic the irregularities found in nature.

“Applied mathematics had been concentrating for a century on phenomena which were smooth, but many things were not like that: the more you blew them up with a microscope the more complexity you found,” said David Mumford, a professor of mathematics at Brown University. “He was one of the primary people who realized these were legitimate objects of study.”

In a seminal book, “The Fractal Geometry of Nature,” published in 1982, Dr. Mandelbrot defended mathematical objects that he said others had dismissed as “monstrous” and “pathological.” Using fractal geometry, he argued, the complex outlines of clouds and coastlines, once considered unmeasurable, could now “be approached in rigorous and vigorous quantitative fashion.”

For most of his career, Dr. Mandelbrot had a reputation as an outsider to the mathematical establishment. From his perch as a researcher for I.B.M. in New York, where he worked for decades before accepting a position at Yale University, he noticed patterns that other researchers may have overlooked in their own data, then often swooped in to collaborate.

“He knew everybody, with interests going off in every possible direction,” Professor Mumford said. “Every time he gave a talk, it was about something different.”

Dr. Mandelbrot traced his work on fractals to a question he first encountered as a young researcher: how long is the coast of Britain? The answer, he was surprised to discover, depends on how closely one looks. On a map an island may appear smooth, but zooming in will reveal jagged edges that add up to a longer coast. Zooming in further will reveal even more coastline.

“Here is a question, a staple of grade-school geometry that, if you think about it, is impossible,” Dr. Mandelbrot told The New York Times earlier this year in an interview. “The length of the coastline, in a sense, is infinite.”

In the 1950s, Dr. Mandelbrot proposed a simple but radical way to quantify the crookedness of such an object by assigning it a “fractal dimension,” an insight that has proved useful well beyond the field of cartography.

Over nearly seven decades, working with dozens of scientists, Dr. Mandelbrot contributed to the fields of geology, medicine, cosmology and engineering. He used the geometry of fractals to explain how galaxies cluster, how wheat prices change over time and how mammalian brains fold as they grow, among other phenomena.

His influence has also been felt within the field of geometry, where he was one of the first to use computer graphics to study mathematical objects like the Mandelbrot set, which was named in his honor.

“I decided to go into fields where mathematicians would never go because the problems were badly stated,” Dr. Mandelbrot said. “I have played a strange role that none of my students dare to take.”

Benoît B. Mandelbrot (he added the middle initial himself, though it does not stand for a middle name) was born on Nov. 20, 1924, to a Lithuanian Jewish family in Warsaw. In 1936 his family fled the Nazis, first to Paris and then to the south of France, where he tended horses and fixed tools.

After the war he enrolled in the École Polytechnique in Paris, where his sharp eye compensated for a lack of conventional education. His career soon spanned the Atlantic. He earned a master’s degree in aeronautics at the California Institute of Technology, returned to Paris for his doctorate in mathematics in 1952, then went on to the Institute for Advanced Study in Princeton, N.J., for a postdoctoral degree under the mathematician John von Neumann.

After several years spent largely at the Centre National de la Recherche Scientifique in Paris, Dr. Mandelbrot was hired by I.B.M. in 1958 to work at the Thomas J. Watson Research Center in Yorktown Heights, N.Y. Although he worked frequently with academic researchers and served as a visiting professor at Harvard and the Massachusetts Institute of Technology, it was not until 1987 that he began to teach at Yale, where he earned tenure in 1999.

Dr. Mandelbrot received more than 15 honorary doctorates and served on the board of many scientific journals, as well as the Mandelbrot Foundation for Fractals. Instead of rigorously proving his insights in each field, he said he preferred to “stimulate the field by making bold and crazy conjectures” — and then move on before his claims had been verified. This habit earned him some skepticism in mathematical circles.

“He doesn’t spend months or years proving what he has observed,” said Heinz-Otto Peitgen, a professor of mathematics and biomedical sciences at the University of Bremen. And for that, he said, Dr. Mandelbrot “has received quite a bit of criticism.”

“But if we talk about impact inside mathematics, and applications in the sciences,” Professor Peitgen said, “he is one of the most important figures of the last 50 years.”

Besides his wife, Dr. Mandelbrot is survived by two sons, Laurent, of Paris, and Didier, of Newton, Mass., and three grandchildren.

When asked to look back on his career, Dr. Mandelbrot compared his own trajectory to the rough outlines of clouds and coastlines that drew him into the study of fractals in the 1950s.

“If you take the beginning and the end, I have had a conventional career,” he said, referring to his prestigious appointments in Paris and at Yale. “But it was not a straight line between the beginning and the end. It was a very crooked line.”
I tagged this topic due to a recent development

Quote:The Associated Press
Saturday, October 16, 2010; 3:25 PM

CAMBRIDGE, Mass. -- Benoit Mandelbrot (ben-WAH' MAN'-dul-braht), a well-known mathematician who was largely responsible for developing the field of fractal geometry, has died. He was 85.

His wife, Aliette, says he died Thursday of pancreatic cancer. He had lived in Cambridge, Mass.

The Polish-born French mathematician founded the field of fractal geometry, the first broad attempt to quantitatively investigate the notion of roughness. He was interested in both the development and application of fractals, which he also showed could be used elsewhere in nature.

For years, he worked for IBM in New York. Later he became Sterling Professor Emeritus of Mathematical Sciences at Yale University.

Mandelbrot also received honorary doctorates and served on boards of scientific journals
He is survived by his wife, two sons and three grandchildren.

The task is to carry on Mandelbrot's work into ever more unknown territories.
I still have the 1985 Scientific American issue that headlined the Mandelbrot Set.
What a revelation!!!
Ever since then, computing the M-set faster/better has become a technology benchmark.'s a series of Lyapunov zoom in/out images with my original placed in context......


[Image: Lyap6zout2.jpg]
[Image: Lyap6zout1.jpg]
[Image: Lyap6.jpg]
[Image: Lyap6zin1.jpg]
[Image: Lyap6zin2.jpg]
[Image: Lyap6zin3.jpg]


I hope this begins to adress your concerns that an artificially introduced tessellation is involved.
The tiling effect seems to be intrinsic to the formula.
Further research may reveal an infinitely microscopic dimension of Escher rotations.
God Bless Sir Mandelbrot

Quote:“I decided to go into fields where mathematicians would never go,
because the problems were badly stated,”
Dr. Mandelbrot said.
“I have played a strange role that none of my students dare to take.”

Thanks Val and Keith for the Mandelbrot info.

Quote:I hope this begins to adress your concerns
that an artificially introduced tessellation is involved

not at all, no concerns, but I am still confused with what you started with.
Where does the image originate, it that beginning image of the zoom sequence?

It makes sense that a Lyapunov fractal regenerates itself infinitely
as that is basically what all fractals do in a way.

Looking at the pattern, it at first appeared like it has flaws in the formula,
as I was focusing on the black swirls that loop around,
and it looked like parts of the pattern had breaks in the black swirl,
but then I realized that the "break"
was actually the black loop plunging through the gold section and submerging
then emerging back out the other side.

This may be a 2D defect in the image.... it is an odd pattern.

I have one very rough set of geometries for you in there of my screen sizes.
In one of the larger images:
Simply I measured with a ruler between the centers of the bright squares.
It is almost dead on 8 cm in a straight horizontal line
between square centers.
Then I took the midpoint on that connecting line,
and measured straight down to the next lower bright square.
{as that is how the squares are layed out geometrically}
It is 4.4 or 4.5 cm along that vertical drop line.

at 4.5 you arctan 8 / 9
at 4.4 you get arctan 10 / 11

So if you can get a better distinct measurement as I have shown,
you at least have one of the replicating geometries.

now the manta ray looking forms have a distinct little X like feature right at the tops.
The mantas form a square-ish polygon in a set of 4 manta's.
See it?
Two manta heads point northeast,
two heads point southeast,

I measured distances between X's....odd....5 cm on the spot between two of them
and about 7 cm part on the other two.
There appears to be a spot on 54 or 55 degree angle in there.
Problem with that:

Bent Pyramid lower half: arctan 1.4 = 54.462 degrees
arctan tetrahedral sqrt 2 = 54.7356 degrees
arctan sqrt phi = 54 degrees.

There is a perfect rt angle in each bright square...
In super zoom
look at the black rt angle that crosses the lower section of each gold bright square.
The angle,
from that black corner,
through the northwest corner of the bright square is 39-40 degrees.
Again two great geometries are possible.
40 degrees means it has some enneagon geometry.
39 degrees means it has Petrie style Menkaure pyramid 39-51 degree geometry
and I have modern phi and Pi equations for that.

Try some measurements from your end.

<img src="{SMILIES_PATH}/cheers.gif" alt="Cheers" title="cheers" />

<img src="{SMILIES_PATH}/applause.gif" alt="Applause" title="applause" />
Quote:...I am still confused with what you started with.
Where does the image originate, it that beginning image of the zoom sequence?

OK...the original image was somewhere around the bottom frame of the series of 5 above.
Look again at this image from my link...

Quote:The diagram below is a zoom in Lyapunov space for the sequence {ab}. The first snapshot corresponds to the window with (0, 0) in the upper left hand corner and (4, 4) in the lower right. The black regions below and to the right are chaotic while the largest black stripe is the family of superstable points. The center of the "x" is the point (2, 2). Other stripes correspond to families of superstable n-cycles. The flotsam in the chaotic regime corresponds to stable islands of odd-multiple periodicity. Again, note the self-referential nature of the islands to the whole picture. The blob in the last picture has the same swallow-tail configuration as the entire space as well as its own atoll of stable islets.

[Image: lyapunov-strip.gif]

See the "manta" at the far right???
That was about the zoom level I started at.
If you keep zooming out to the left, then you get to the tiling.

If you want, then I can post the Lyapunov module list, so you can inspect the variables.
The Lyapunov formula I was using is... a*sin(x+r)^2
Variable a, for instance, is apparently the "a" in the sequence parameter. As yet, I'm not sure what r represents, unless it stands for radius.
There are 4 other formulas, but this one really delivered the complex Escher effect. look again at this image from the link... ...

[Image: lyap.4.01.png]

You can see the origin of the "manta" as well as the right angle effects.

Here's what happened when I changed the sequence {ab} to {aab}...

[Image: Lyapunov7b.jpg]

No, I don't understand how the two sequence variables interact before a plotpixel command...maybe a quadratic equation???, I'm somewhat hesitant about posting this next image.
There is another parameter "A", which all I can figure out is a real number input.
I changed the default 2 to 11 and came up with this...
On my monitor I see bands of precisely plotted tiny random(?) patterns, almost like a bar code. If I zoom into those bands then all I see are ever more finely resolved bands until at some unimaginable decimal level I eventually reach a point where the program resolution simply gives out into big blocks.
The criticism might be raised that... 
Whistle I'm not saying this doesn't represent a Contact/Secret Alien encoding of knowledge far beyond what we know to be reality......

[Image: Lyapunov7a.jpg]

[scroll down this page... ... _space.php]
Fascinating stuff Kalter.
Great ideas by the ways.
I will need a few days to catch up.

Quote:I'm not saying this doesn't represent a Contact/Secret Alien encoding of knowledge
far beyond what we know to be reality......

It's like the DMT isn't it?
Or more...
All Dimensional language.
Lyapunov, quaternion or common fractal.
In my earlier electric light grids, I called them Pleiadean light-sound communication signals,
to fancy up the image title.
When too high on the DMT's one could even....taste and smell the light-sound communication.

Which comes to a genetic factor over 50000 years.
The indoles or ancient soma which is almost certainly psilocybe DMT,
impart the visionary aspects of infinite replicating grids to psychedelic E8 formats,
to quantum fractal imagery,
that inevitably influences the human mind over thousands of generations,
and IMO affects as well the evolution of mind and pineal originated all intuitive perceptions,
which are inevitably tied to communications without words.

This is one of the reasons big Pharma plugs your pineal and brain with toxic drugs,
and why fluoride is in toothpaste and water supplies,
it plugs the pineal with....I think it was calcium hydroxyapatite crystals ...
see below

So imagine a gland in more advanced alien species,
that can create the "think speak" audiovisual clairvoyance from fractal patternings.

Little too sci fi perhaps.

But I often wondered if psychedelic based spore is an implant from... somewhere and something else,
after all,
there is no question, everybody that partakes in DMT indoles,
knows that there is a complete universe there overlapping ours,
and as such that molecule is a transdimensional bridge in a fashion.

Also to consider is that the most potent strain here is Cyanescens
and are almost exclusively found in mulches and chipped wood debris from man's activity,

title : Fluoride Deposition in the Aged Human Pineal Gland Author(s): J. Luke
Info : Figures: 2; Tables: 0; References: 32 Keywords : Calcium; Distribution; Fluoride; Human pineal gland; Hydroxyapatite; Pineal concretions Abstract : The purpose was to discover whether fluoride (F) accumulates in the aged human pineal gland. The aims were to determine (a) F-concentrations of the pineal gland (wet), corresponding muscle (wet) and bone (ash); (b) calcium-concentration of the pineal. Pineal, muscle and bone were dissected from 11 aged cadavers and assayed for F using the HMDS-facilitated diffusion, F-ion-specific electrode method. Pineal calcium was determined using atomic absorption spectroscopy. Pineal and muscle contained 297+/-257 and 0.5+/-0.4 mg F/kg wet weight, respectively; bone contained 2,037+/-1,095 mg F/kg ash weight. The pineal contained 16,000+/-11,070 mg Ca/kg wet weight. There was a positive correlation between pineal F and pineal Ca (r = 0.73, p<0.02) but no correlation between pineal F and bone F. By old age, the pineal gland has readily accumulated F and its F/Ca ratio is higher than bone.
A great restriction (so far) with these Lyapunov "tilings" seems to be the apparently intrinsic quadrilateral form......
...although this site (and you as well?) says that Phi is the key to pentagonal geometry......

All else being equal......

[Image: Lyapunov6phi-1.jpg]

[Image: Lyapunov6pi.jpg]
Pentagonal geometry is composed of Phi.

phi angles:

18, 36, 54, 72

sine 54 degrees = Phi / 2

cosine 72 x Phi squared = sine 54 = phi / 2

tangent 72 / by Phi = sqrt 3.618033989

one thing I created was universal harmonic pi = uPi.

Phi squared / by uPi = 5 / 6 = 0.8333333333~

one can actually find uPi within square root 2,3 and 5 geometries and/or associated math.

look at
Sqrt. 5

arctangent sqrt. 5 = 65.90515745 <-----

sine of 65.90515745 degrees = Sqrt of 0.83333333~ or sqrt. { 5 / 6}

here is a tetrahedral and phi geometry pyramid

notice angle 2e

[Image: U31lC.jpg]

I like your ideas of using constants a lot, or combinations of them,
and no doubt the usage of uPi or 3.141640787
will probably produce very little difference in the A = pi image.

you might try Phi squared and inverse phi
for A,
square root 2 ...and 3....and 5,
definitely square root two, because that has the replicating square root two rectangle.

Also 3 constants are wrapped in these values:
and 1.87

to be exact:
Pi x phi / e = 1.870007392

phi x e / pi = 1.400012642

I do have a request though:
sqrt 10 phi
4.022479321 <------ = A

just to see .... if there are any visual somethings to see.'s two rough renditions of sqrt 10 phi...A=4.022479321......
the difference being the Lyapunov formula used......

[Image: Lyap6sqrt10phi.jpg]

[Image: Lyap6sqrt10phi2.jpg]
I'm trying to find a way to get around the global coloration,
but this rendering of the same sqrt(10Phi) parameters is ...interesting......

[Image: Lyap6invmap.jpg]
[Image: Lyap6invmap4.jpg]
Holy fuck Damned Smoke Tan

The last two images are fabulous!

I have to work all day today on finishing pdf 2 in my Teotihuacan grids expose.

When that is done,
I can start to spend more time here at this.
I have been required to do a lot of traveling lately as well,
with two 200 mile round trips to Seattle and back in 3 days.
But I will try and get back to this soon.

Those sqrt 10 Phi images are superb.
Man they really popped out a cool visual!

Great work Kalter.
How do the initial images change from standard to that Lyapunov Big Bang?

That is just too cool.

It's fractal psychedelic .......fractadelic.......psyfractadelic ....mannnnnnnnnnn

Well sqrt 10 phi is a winner there,
I had hoped so, it is used in some of my calculations.
It is a bit sophisticated of a value.

If you want, you could try Egyptian style tangents.

22 / 14 and 63 / 99
but those might be too standard.

there are a ton of "Convergence Dynamics" math values that could be tested.


10 inverse Phi = 6.18033989
I was observing  this because it has a convergence to the number 61

10 inverse Phi x Pi squared = 60.9975
the Pi value to make that exact 61 is 3.141656781

<img src="{SMILIES_PATH}/hmm2.gif" alt="Hmm2" title="hmm2" />

There are a ton of constants
e = 2.71828 ... you might try that just to make sure...
planck's coefficient = 6.626068
there's the Euler Masheroni constant...

planck x tangent 54 = 9.12 ---------- 9.120001575.
912 / 399 jupiter synod = 16 / 7

then there are combos and two of these probably should be tested at some point


1.4 = Bent Pyramid lower half tangent = Phi x e / by Pi -- 1.400012642

1.87 = Pi x Phi / e = 1.870007392

it is endless ... the possibilities.....swept away by infinity.....swooooshhhh shhh shh .... ... .. .

there are the bridge railings in the upper corners

[Image: aCq88.jpg]
Thank you, V!!!

Quote:How do the initial images change from standard to that Lyapunov Big Bang?

You might want to read about the geometry behind my last 2 images.

Quote: [Image: Inversion_800.gif]
The property that inversion transforms circles and lines to circles or lines (and that inversion is conformal) makes it an extremely important tool of plane analytic geometry. By picking a suitable inversion circle, it is often possible to transform one geometric configuration into another simpler one in which a proof is more easily effected. The illustration above shows examples of the results of geometric inversion.

Look here... ... xplanation ...for an applet that demonstrates inversion.
At the end of the page is a list of related circle transformations/applets.

Here's an inversion of a recent image...

[Image: Lyap6invmap5.jpg]
So far, I haven't figured out how to do inversions of 3D Mandelbrots......
I cannot say that I understand that,
but I will try to soon.
It is none the less fascinating to see that set of diagrams from

I think that it is at Wolfram
that they have the link on
Euler Masheroni constant

what a fucked up ...thang...that is.
It is like a math constant for thong...

<img src="{SMILIES_PATH}/rofl.gif" alt="Rofl" title="rofl" />

it is as bad or worse than Pi with infinite decimal

at wolfram they challenge people to come up with a formula
to see how many placements of the Euler Masheroni constant you can create.

I didn't do too badly actually,
and beat many of the selections I have seen posted there and elsewhere.
Some of the equations the mathematicians came up with were wild.

I used a quick 30 minute Harmonic Code search
of course
simplified modern mathematics

divided by:
{Sqrt 0.1 + 1} squared = masheroni constant

do the bottom equation first,
then simply use 1/x on your calculator

if you don't have 1/x, go buy a 10$ TI 30XA

....I think I got the first 6 decimal placements
which is fairly good for a 30 minute try,
using only the number 1


Return to the Wolfram site...
Look at this pair of diagrams...

Quote:[Image: Checker_700.gif] [Image: InverseChecker_700.gif]

The above plot shows a chessboard centered at (0, 0) and its inverse about a small circle also centered at (0, 0) (Gardner 1984, pp. 244-245; Dixon 1991).

<img src="{SMILIES_PATH}/hmm2.gif" alt="Hmm2" title="hmm2" /> near as I can figure...
The points in the chessboard, when "mapped" to a circle of a given radius, are displaced according to this diagram...

Quote:[Image: InversePoints_701.gif]

Replace line L with the chessboard.
The points in the chessboard are shifted  such that point P maintains its position as an endpoint in a series of right triangles. The result is that the chessboard is warped into concentric intersecting circles.


Now...what happens when the chessboard is replaced by a fractal curve???

Look at this earlier Lyapunov tessellation and think of it as the chessboard......

[Image: Lyap6zout1.jpg]

Next is the zoomed-out inversion...

[Image: Lyap6invmap6.jpg]

...and here is the zoomed-in inversion...

[Image: Lyap6invmap7.jpg]
From the Infinite Abyss of Absolute Nothingness,
the point source of empty pointlessness....
Quantum Foam
in fractal dynamic expansions.

Great work Kalter.
You are on to something deep here.
like the infinite abyss
Nothingness generates

keep going.

<img src="{SMILIES_PATH}/cheers.gif" alt="Cheers" title="cheers" />
My image program fails me in this way:

I can create a giant single master hexagon or pentagon or enneagon grid
like 5000 by 5000 pixels,
I cannot produce a packed grid of interconnnected hexes or pentagons etc.
once the master pentagon of hex is created
all my program will allow me to do is make a ring of pentagons. ...etc

So I did this finally.

I had to reduce the image size by a factor of 5
to get the ring to fit ion a 5000 by 5000 grid

what you get with a pentagon grid ring
is a ten pointed star with a central empty space large decagon.

Cool but no cigar.

Well I am determined to salvage some form of unique geometry
with grid replications
and I got one to pop.

Check it out, this is in pure Phi geometry all throughout.
I got an 8 pointed star
with 16 sides,
with a central hexagon encompassed empty space
centered by a parallelogram.....
in pentagonal geometry

The outer ring
is two sets of 3 pentagons with a parallelogram at top and bottom as the juncture
between the two sets of pentagons.
That upper and lower parallelogram have two 72 and two 108 degree angles.
The central hex has two 72 and four 144 degree angles.
the central parallelogram has two 36 and two 144 degree angles.

All phi geometry.

The the close up from the interior of the lyapunov master pentagon grid
and they called it a Markus-Lyapunov fractal.

Now the close up....
inside there are a couple  7 sided geometries.
they are tough to find, but they are there which is absolutely wild.

8 pointed star with 16 sides and central hex with parallelogram
all in phi geometry
[Image: 4jQx5.jpg]

close up interior master grid
[Image: NIEhl.jpg]
I went out this morning to pick Birch Boletes in my neighbor's yard,
because they have popped up again with the sudden rain we finally got.
And lo and behold...
Lyapunov fractal shrooms were there as well!
They look to be growing in Birch mulch, and if so,
this will be a first in shroomology!
There are no there trees there, and no alders anywhere near.
Unless a big piece of alder is underground....which is unlikely,
they are growing on Birch mulch.
If so, next year they may be everywhere there.
Too bad I don;t eat the Lyapunov's but hey, check it out!
Cyans growing in birch mulch in grass.
Top left are the baby Birch Boletes,.
bottom right are the big boys in front of 120 pound boulder
of black nephrite jade that I am going to work very hard on.
Right now I am attempting to clean off the bio-plaque coating,
and that top surface is now 75% cleaned.

[Image: qJGyb.jpg]
Quote:Vianova says...
The the close up from the interior of the lyapunov master pentagon grid
and they called it a Markus-Lyapunov fractal.

Hmmm...I added that to searching for "Lyapunov tesselations" and found the original here... http://charles.vassallo.pagesperso-oran ... omp13.html ...the other examples are also pretty interesting......
I still haven't found anything else that looks like the escheresque designs I've posted, although such tessellations are the subject of a lot of abstract papers, but no sample images. I have to think they're talking about the same kind of patterns.
One thing that happens is that the escher grid appears in the normal plot when zoomed OUT, but in the inversion rendition it appears when zoomed IN.
<img src="{SMILIES_PATH}/dunno.gif" alt="Dunno" title="dunno" /> ...maybe on some level that's a no-brainer, but that's only because I can't display the entire Lyapunov formula in the editor to see what's going on......

I had deleted this next a while back because of all the black background...whatever...
and I didn't like the very top whitish bands...etc, etc......

[Image: Mand3D136.jpg]
[Image: Mand3D150.jpg]
[Image: Mand3D151.jpg]
Here is a Lyapunov inversion mapped to the Weierstrass/Sigma function......

[Image: LyapinvwWeir1.jpg]